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36
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 476 (123 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
A Linear Upper Bound in Extremal Theory of Sequences
, 1994
"... An extremal problem considering sequences related to DavenportSchinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) ..."
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Cited by 49 (14 self)
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An extremal problem considering sequences related to DavenportSchinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) a r = a s ; r 6= s implies jr \Gamma sj k and 3) u contains no subsequence of the type x k (x stands for xx::x itimes).
Improved bounds and new techniques for DavenportSchinzel sequences and their generalizations
 In Proceedings 20th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2009
"... We present several new results regarding λs(n), the maximum length of a Davenport–Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n · 2 (1/t!)α(n)t +O(α(n) t−1), n · 2 (1/t!)α(n)t log 2 α(n)+O(α(n) t), s ≥ 4 even; s ≥ 3 odd; where t = ⌊(s − 2)/2⌋, and α(n) denotes th ..."
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Cited by 34 (2 self)
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We present several new results regarding λs(n), the maximum length of a Davenport–Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n · 2 (1/t!)α(n)t +O(α(n) t−1), n · 2 (1/t!)α(n)t log 2 α(n)+O(α(n) t), s ≥ 4 even; s ≥ 3 odd; where t = ⌊(s − 2)/2⌋, and α(n) denotes the inverse Ackermann function. The previous upper bounds, by Agarwal, Sharir, and Shor (1989), had a leading coefficient of 1 instead of 1/t! in the exponent. The bounds for even s are now tight up to lowerorder terms in the exponent. These new bounds result from a small improvement on the technique of Agarwal et al. More importantly, we also present a new technique for deriving upper bounds for λs(n). This new technique is based on some recurrences very similar to those used by the author, together with Alon, Kaplan, Sharir, and Smorodinsky (SODA 2008), for the problem of stabbing interval chains with jtuples. With this new technique we: (1) rederive the upper bound of λ3(n) ≤ 2nα(n)+O ( n √ α(n) ) (first shown by Klazar, 1999); (2) rederive our own new upper bounds for general s; and (3) obtain improved upper bounds for the generalized Davenport–Schinzel sequences considered by Adamec, Klazar, and Valtr (1992). Regarding lower bounds, we show that λ3(n) ≥ 2nα(n) − O(n) (the previous lower bound (Sharir and Agarwal, 1995) had a coefficient of 1 2), so the coefficient 2 is tight. We also present a simpler variant of the construction of Agarwal, Sharir, and Shor that achieves the known lower bounds of λs(n) ≥ n·2 (1/t!)α(n)t−O(α(n) t−1) for s ≥ 4 even.
Graph Drawings With No K Pairwise Crossing Edges
, 1997
"... A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . It is known that, for any fixed k, any geometric graph G on n vertices with no k pairwise crossing ed ..."
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Cited by 32 (1 self)
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A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . It is known that, for any fixed k, any geometric graph G on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. In this paper we give a new, simpler proof of this bound, and show that the same bound holds also when the edges of G are represented by xmonotone curves (Jordan arcs). 1 Introduction A graph drawing is a drawing of a graph in the plane such that each vertex is represented by a distinct point and each edge is represented by a Jordan arc connecting the corresponding two points (vertices) so that any two arcs meet in at most one point which is either a common endpoint or a common interior point where the two arcs cross each other. A geometric graph is a graph drawing in which all arcs are straight line segments. See [8] for a survey of results about ...
On Geometric Graphs With No K Pairwise Parallel Edges
 Discrete Comput. Geom
, 1997
"... A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . Two edges of a geometric graph are said to be parallel , if they are opposite sides of a convex quadr ..."
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Cited by 28 (1 self)
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A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . Two edges of a geometric graph are said to be parallel , if they are opposite sides of a convex quadrilateral. In this paper we show that, for any fixed k 3, any geometric graph on n vertices with no k pairwise parallel edges contains at most O(n) edges, and any geometric graph on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. We also prove a conjecture of Kupitz that any geometric graph on n vertices with no pair of parallel edges contains at most 2n \Gamma 2 edges. 1 Introduction A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . See [9] for a survey of results about geometric graphs. Two edges of a geomet...
Generalized DavenportSchinzel Sequences
, 1993
"... The extremal function Ex(u; n) (introduced in the theory of DavenportSchinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa : : : the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following ..."
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Cited by 26 (4 self)
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The extremal function Ex(u; n) (introduced in the theory of DavenportSchinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa : : : the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following the idea of J. Nesetril) we generalize this concept for arbitrary sequence u. We summarize the already known properties of Ex(u; n) and we present also two new theorems which give good upper bounds on Ex(u; n) for u consisting of (two) smaller subsequences u i provided we have good upper bounds on Ex(u i ; n). We use these theorems to describe a wide class of sequences u ("linear sequences") for which Ex(u; n) = O(n). Both theorems are used for obtaining new superlinear upper bounds as well. We partially characterize linear sequences over three symbols. We also present several problems about Ex(u; n).
Splay trees, DavenportSchinzel sequences, and the deque conjecture
, 2007
"... We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct ..."
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Cited by 21 (6 self)
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We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan’s deque conjecture and prove that n deque operations require O(nα ∗ (n)) time, where α ∗ (n) is the minimum number of applications of the inverseAckermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.
Generalized DavenportSchinzel sequences: results, problems, and applications
, 1994
"... We survey in detail... ..."
On the Maximum Lengths of DavenportSchinzel Sequences
 Contemporary Trends in Discrete Mathematics, ˇ Stiˇrín Castle 1997 (Czech Republic), American Mathematical Society, Providence RI
, 1999
"... The quantity N5 (n) is the maximum length of a finite sequence over n symbols which has no two identical consecutive elements and no 5term alternating subsequence. Improving the constant factor in the previous bounds of Hart and Sharir, and of Sharir and Agarwal, we prove that N5 (n) ! 2nff(n) ..."
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Cited by 18 (3 self)
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The quantity N5 (n) is the maximum length of a finite sequence over n symbols which has no two identical consecutive elements and no 5term alternating subsequence. Improving the constant factor in the previous bounds of Hart and Sharir, and of Sharir and Agarwal, we prove that N5 (n) ! 2nff(n) +O(nff(n) ); where ff(n) is the inverse to the Ackermann function. Quantities Ns (n) can be generalized and any finite sequence, not just an alternating one, can be assigned extremal function. We present a sequence with no 5term alternating subsequence and with an extremal function AE n2 .
Generalized DavenportSchinzel Sequences and Their 01 Matrix Counterparts
, 2010
"... A generalized DavenportSchinzel sequence is one over a finite alphabet whose subsequences are not isomorphic to a forbidden subsequence σ. What is the maximum length of such a σfree sequence, as a function of its alphabet size n? is the extremal function linear or nonlinear? and what characteristi ..."
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Cited by 11 (3 self)
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A generalized DavenportSchinzel sequence is one over a finite alphabet whose subsequences are not isomorphic to a forbidden subsequence σ. What is the maximum length of such a σfree sequence, as a function of its alphabet size n? is the extremal function linear or nonlinear? and what characteristics of σ determine the answers to these questions? It is known that such sequences have length at most n · 2 (α(n))O(1), where α is the inverseAckermann function and the O(1) depends on σ. We resolve a number of open problems on the extremal properties of generalized DavenportSchinzel sequences. Among our results: 1. We give a nearly complete characterization of linear and nonlinear σ ∈ {a, b, c} ∗ over a threeletter alphabet. Specifically, the only repetitionfree minimally nonlinear forbidden sequences are ababa and abcacbc. 2. We prove there are at least four minimally nonlinear forbidden sequences. 3. We prove that in many cases, doubling a forbidden sequence has no significant affect its extremal function. For example, Nivasch’s upper bounds on alternating sequences of the form (ab) t and (ab) t a, for t ≥ 3, can be extended to forbidden sequences of the form (aabb) t and (aabb) t a. 4. Finally, we show that the absence of simple subsequences in σ tells us nothing about σ’s extremal function. For example, for any t, there exists a σt avoiding ababa whose extremal function is Ω(n·2 αt (n) Most of our results are obtained by translating questions about generalized DavenportSchinzel sequences into questions about the density of 01 matrices avoiding certain forbidden submatrices. We give new and often tight bounds on the extremal functions of numerous forbidden 01 matrices.